The study is to evaluate the effects of a synthetic human growth hormone (rhGH) on children who have had renal transplants. It will assess the following: whether treatment with rhGH will result in improved growth, whether and how treatment with rhGH affects acute and chronic rejection, common problems in children who have had a renal transplant, whether and how treatment with rhGH affects the functioning of the kidney transplant, and whether treatment with rhGH results in any significant side effects. Patients will be randomized to receive rhGH immediately or be in a control group to receive rhGH after one year of randomization. A total of 100 patients in each group will be enrolled throughout the nation. All patients will receive two years of study medication.